Kinetic Theory Derivations with Figures & Animation by Academic Torch

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Kinetic Theory of Gases — Physics

Kinetic Theory of Gases

Physics • Molecular Theory of Gases

The kinetic theory of gases connects the microscopic motion of molecules with the macroscopic properties of gases. Let’s derive its key results step by step.

1. Molecule in a Cube

Imagine a cube of side L containing a single gas molecule of mass m. Its velocity has components \(v_x, v_y, v_z\).

A molecule colliding with cube walls.

2. Momentum Change

Δp = -2 m vₓ

On colliding with a wall perpendicular to the x-axis, momentum reverses: from \(+mv_x\) to \(-mv_x\). Hence, change = \(-2mv_x\). This momentum is transferred to the wall.

3. Time Between Collisions

Δt = 2L / vₓ

The molecule must travel to the opposite wall and back before colliding again. Distance = \(2L\), speed = \(v_x\), so time = \(2L / v_x\).

4. Average Force

F = m vₓ² / L

Force = (change in momentum) ÷ (time interval). Substituting: \(F = (2mv_x) / (2L/v_x) = m v_x^2 / L\).

5. Pressure from One Molecule

P = m vₓ² / V

Pressure = Force ÷ Area. Area of wall = \(L^2\). Thus, \(P = (m v_x^2 / L) ÷ L^2 = m v_x^2 / V\).

6. Total Pressure from N Molecules

P = (N m ⟨vₓ²⟩) / V

For N molecules, we sum their contributions. Using average square velocity: \(⟨v_x^2⟩\).

7. Velocity Components

⟨v²⟩ = 3⟨vₓ²⟩ → ⟨vₓ²⟩ = ⅓⟨v²⟩

Since motion is random and isotropic, energy is equally distributed among x, y, z directions. So the mean square speed along one axis is one-third of the total.

8. Final Pressure Formula

P = ⅓ (Nm/V) ⟨v²⟩ = ⅓ ρ vᵣₘₛ²

Let \(\rho = Nm/V\) (density), and define root mean square speed \(v_{rms} = \sqrt{⟨v²⟩}\). Then: \(P = \tfrac{1}{3}\rho v_{rms}^2\).

9. RMS Speed Expression

vᵣₘₛ = √(3RT/M)

Relating kinetic energy to temperature: \(\tfrac{1}{2}m⟨v^2⟩ = \tfrac{3}{2}kT\). In molar form, \(v_{rms} = \sqrt{3RT/M}\).

DO YOU KNOW?
The average kinetic energy per molecule is Eₖ(avg) = 3/2 kT, independent of the type of gas.

10. Examples

Example 1: RMS speed of H₂ at 300 K ≈ 1900 m/s.
Example 2: Ratio of speeds v(H₂)/v(O₂) = √(M(O₂)/M(H₂)) = 4.

Interactive Demo: Effect of Temperature

Adjust the temperature to see how RMS speed changes.

RMS Speed of O₂: 0 m/s

Key Takeaways

  • Gas pressure arises from collisions of molecules with container walls.
  • Temperature is a direct measure of average kinetic energy.
  • The Kinetic Theory explains macroscopic gas laws using microscopic motion.

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